Displaying similar documents to “Prime numbers with Beatty sequences”

Carmichael numbers composed of primes from a Beatty sequence

William D. Banks, Aaron M. Yeager (2011)

Colloquium Mathematicae

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Let α,β ∈ ℝ be fixed with α > 1, and suppose that α is irrational and of finite type. We show that there are infinitely many Carmichael numbers composed solely of primes from the non-homogeneous Beatty sequence ℬ α , β = ( ⌊ α n + β ⌋ ) n = 1 ∞ . We conjecture that the same result holds true when α is an irrational number of infinite type.

The binary Goldbach conjecture with primes in arithmetic progressions with large modulus

Claus Bauer, Yonghui Wang (2013)

Acta Arithmetica

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It is proved that for almost all prime numbers k ≤ N 1 / 4 - ϵ , any fixed integer b₂, (b₂,k) = 1, and almost all integers b₁, 1 ≤ b₁ ≤ k, (b₁,k) = 1, almost all integers n satisfying n ≡ b₁ + b₂ (mod k) can be written as the sum of two primes p₁ and p₂ satisfying p i ≡ b i ( m o d k ) , i = 1,2. For the proof of this result, new estimates for exponential sums over primes in arithmetic progressions are derived.

On the Brun-Titchmarsh theorem

James Maynard (2013)

Acta Arithmetica

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The Brun-Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq ≥ 8 and q is sufficiently large. Moreover, we also produce a lower bound of size x / ( q 1 / 2 ϕ ( q ) ) when logx/logq ≥ 8 and...

On congruent primes and class numbers of imaginary quadratic fields

Nils Bruin, Brett Hemenway (2013)

Acta Arithmetica

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We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate-Shafarevich groups of the associated elliptic curves. We also present a related criterion for primes p such that 16 divides the class number of the imaginary quadratic field...

Truncatable primes and unavoidable sets of divisors

Artūras Dubickas (2006)

Acta Mathematica Universitatis Ostraviensis

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We are interested whether there is a nonnegative integer u 0 and an infinite sequence of digits u 1 , u 2 , u 3 , ⋯ in base b such that the numbers u 0 b n + u 1 b n - 1 + ⋯ + u n - 1 b + u n , where n = 0 , 1 , 2 , ⋯ , are all prime or at least do not have prime divisors in a finite set of prime numbers S . If any such sequence contains infinitely many elements divisible by at least one prime number p ∈ S , then we call the set S unavoidable with respect to b . It was proved earlier that unavoidable sets in base b exist if b ∈ { 2 , 3 , 4 , 6 } , and that no unavoidable set exists in base b = 5 . Now,...

Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields

Hans Roskam (2001)

Journal de théorie des nombres de Bordeaux

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Let S be a linear integer recurrent sequence of order k ≥ 3 , and define P S as the set of primes that divide at least one term of S . We give a heuristic approach to the problem whether P S has a natural density, and prove that part of our heuristics is correct. Under the assumption of a generalization of Artin’s primitive root conjecture, we find that P S has positive lower density for “generic” sequences S . Some numerical examples are included.

Consecutive primes in tuples

William D. Banks, Tristan Freiberg, Caroline L. Turnage-Butterbaugh (2015)

Acta Arithmetica

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In a stunning new advance towards the Prime k-Tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple ( x ) = g x + h j j = 1 k of linear forms in ℤ[x], the set ( n ) = g n + h j j = 1 k contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that ( n ) = g n + h j j = 1 k contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps δ 1 , . . . , δ m form an increasing...

Generalizing a theorem of Schur

Lenny Jones (2014)

Czechoslovak Mathematical Journal

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In a letter written to Landau in 1935, Schur stated that for any integer t > 2 , there are primes p 1 < p 2 < ⋯ < p t such that p 1 + p 2 > p t . In this note, we use the Prime Number Theorem and extend Schur’s result to show that for any integers t ≥ k ≥ 1 and real ϵ > 0 , there exist primes p 1 < p 2 < ⋯ < p t such that p 1 + p 2 + ⋯ + p k > ( k - ϵ ) p t .

On the Piatetski-Shapiro-Vinogradov theorem

Angel Kumchev (1997)

Journal de théorie des nombres de Bordeaux

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In this paper we consider the asymptotic formula for the number of the solutions of the equation p 1 + p 2 + p 3 = N where N is an odd integer and the unknowns p i are prime numbers of the form p i = [ n 1 / γ i ] . We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case γ 1 = γ 2 = γ 3 = γ our theorem implies that every sufficiently large odd integer N may be written as the sum of three Piatetski-Shapiro primes of type γ for 50 / 53 &lt; γ &lt; 1 . ...